We are decapitalizing the contributions' names ...

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / substitution / gdrop.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma
new file mode 100644 (file)
index 0000000..eb4bd48
--- /dev/null
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/gdrop.ma
@@ -0,0 +1,80 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "Basic_2/grammar/genv.ma".
+
+(* GLOBAL ENVIRONMENT SLICING ***********************************************)
+
+inductive gdrop (e:nat): relation genv ≝
+| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆)
+| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G
+| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2
+.
+
+interpretation "global slicing" 
+   'RDrop e G1 G2 = (gdrop e G1 G2).
+
+(* basic inversion lemmas ***************************************************)
+
+lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H >H -H >commutative_plus #H
+  lapply (le_plus_to_le_r … 0 H) -H #H
+  lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ #H2
+  lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+  lapply (lt_plus_to_lt_l … 0 H) -H #H
+  elim (lt_zero_false … H)
+]
+qed-.
+
+lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
+#G1 #G2 #e * -G1 -G2 //
+[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H
+  lapply (le_plus_to_le_r … 0 H) -H #H
+  lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ normalize #H2
+  <(injective_plus_l … H2) in H1; -H2 #H
+  elim (lt_refl_false … H)
+]
+qed-.
+
+fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
+                       e < |G1| → ⇩[e] G1 ≡ G2.
+#I #G #G1 #G2 #V #e * -G -G2
+[ #G #H1 #H destruct #H2
+  lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+  lapply (lt_plus_to_lt_l … 0 H) -H #H
+  elim (lt_zero_false … H)
+| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
+  elim (lt_refl_false … H)
+| #J #G #G2 #W #_ #HG2 #H destruct //
+]
+qed.
+
+lemma gdrop_inv_lt: ∀I,G1,G2,V,e.
+                    ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
+/2 width=5/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
+#e #G1 elim G1 -G1 /3 width=2/
+#I #V #G1 * #G2 #HG12
+elim (lt_or_eq_or_gt e (|G1|)) #He
+[ /3 width=2/
+| destruct /3 width=2/
+| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
+]
+qed.